Efficiency of energy funneling in the photosystem II supercomplex of higher plants.
The investigation of energy transfer properties in photosynthetic multi-protein networks gives insight into their underlying design principles. Here, we discuss excitonic energy transfer mechanisms of the photosystem II (PS-II) CSM supercomplex, which is the largest isolated functional unit of the photosynthetic apparatus of higher plants. Despite the lack of a decisive energy gradient in CSM, we show that the energy transfer is directed by relaxation to low energy states. CSM is not organized to form pathways with strict energetic downhill transfer, which has direct consequences on the transfer efficiency, transfer pathways and transfer limiting steps. The exciton dynamics is sensitive to small structural changes, which, for instance, are induced by the reorganization of vibrational coordinates. In order to incorporate the reorganization process in our numerical simulations, we go beyond rate equations and use the hierarchically coupled equation of motion approach (HEOM). While transfer from the peripherical antenna to the proteins in proximity to the reaction center occurs on a faster time scale, the final step of the energy transfer to the RC core is rather slow, and thus the limiting step in the transfer chain. Our findings suggest that the structure of the PS-II supercomplex guarantees photoprotection rather than optimized efficiency.
Photosynthesis in which light is absorbed and converted in chemical energy is the most important process in nature. In higher plants the light-harvesting machinery is assembled of CSM supercomplexes and networks of LHCII pigment-proteins 1, 2, 3, 4 located in the grana membrane. The CSM supercomplex is formed by a dimeric photosystem II (PS-II) with moderately attached LHCII trimers and several minor complexes 5. Energy transfer to the reaction center (RC) core pigments of PSII, in which the primary step of charge separation initializes an avalanche of photochemical reactions 6, 7, 8, reaches remarkable efficiencies of up to 9. However, it remains unclear of how such high efficiencies can be achieved in large and disordered systems. In contrast to the photosynthetic apparatus of Green Sulfur Bacteria in which fast transfer is guaranteed by efficient energy funneling 10, microscopic derived Hamiltonians do not predict a decisive energy gradient among the individual proteins of the CSM supercomplex 11, 12, 13, 14, 15.
Previous works describe the transfer kinetics with phenomenological models, and extract certain decay components such as the migration time (average time that it takes for an excitation to reach the RC) and trapping time by fitting to fluorescence decay lines 16, 17, 18, 19. Several rate limiting models are discussed in literature 17, 18, 20, 21. Recent studies favor the so called transfer-to-trap limited kinetic model 17, 18, 16, 12 in which the transfer rate from the antenna complexes of PS-II to the RC is proposed to be the transfer limiting step. However, different kinetic models can be fitted equally well to measured fluorescence decay curves 22, and structure based models of energy transfer become necessary to shed light on the underlying transfer mechanisms. First microscopic simulations of the exciton dynamics in the CSM supercomplex show that the overall transfer is driven by a complex interplay of multiple rates rather than through a single transfer-limiting step 22.
In pigment-protein complexes directionality of energy transfer is driven by energy relaxation. Variations in the energy bands of the individual proteins in CSM are not as distinct as in other photosynthetic systems. Nevertheless, the energy gradient in CSM is not completely flat, and the pigments form a certain structure in the energetic layout. For example CP43 and CP47 are lower in energy than the LHCII antenna complexes 14, 11, 12. However, energy transfer in CSM is not a cascade of downhill steps toward the reaction center. Actually the pigments in the proximity of the RC core are the energetically lowest ones 23. Therefore, the last transfer step to the trap needs to overcome an energy barrier which supports the proposed transfer-to-trap limited exciton dynamics in CSM. The transfer limiting step to the RC core pigments, which is not anticipated in previous structure based simulations 22, becomes more evident once we include the recently derived Hamiltonian of CP29 15. The latter is substituted in Ref. 22 by a LHCII monomer. We show that the minor complex CP29 modifies the pathway of energy flow and yields a relaxation channel which drives energy from the peripherical antenna towards pigments closer to PS-II.
The transfer properties are sensitive to small structural modulations which is an immediate consequence induced by the flat energy gradient. There are two major mechanisms which change the energetic structure: (i) static disorder in which site energies are subjected to random fluctuations on much slower time scales than the exciton dynamics and (ii) the reorganization process in which vibrational coordinates relax to a new equilibrium position after a vertical Franck-Condon transition to the excited state energy potential surface 24. During this process the reorganization energy is dissipated in the protein environment. While the transfer times of an ensemble of individual disorder realizations are randomly distributed around some average value 22, the reorganization process is a systematic effect pertaining to the dynamics in all realizations in the same way.
Due to the lack of the computational capability to carry out accurate calculations of the exciton dynamics, previous simulations of transfer time-scales in light-harvesting complexes (LHCs) employ a combined modified Redfield/generalized Förster rate equation approach 25, 26, 27, 22, 28. However the combined modified Redfield/generalized Förster lacks the ability to simulate the reorganization process. In addition those models provide an ad hoc description of dynamic localization, and depend on an empirical cut-off parameter. Recently, a non-Markovian (ZOFE) quantum master equation description is employed to investigate robustness of transfer effiency and the importance of vibrational enhanced transfer in PS-II 29. Here, we perform accurate simulations based on the hierarchically coupled equations of motion approach (HEOM) 30, 31, 32, 33, 34 which accurately incorporates the reorganization process and works for a wide parameter range for the coupling strength to the environment.
Since the computational complexity of HEOM scales exponentially with increasing system size, novel algorithms based on optimized parallelization schemes have been developed 35, 36, 37. The most efficient implementation 35 employs the high compute throughput provided by modern graphics processing units (GPUs) for which a cloud computing version is hosted on nanohub.org 38. GPU-HEOM is bound to the available GPU memory, and simulations are limited to intermediate sized systems. Here we overcome the memory limitation by using QMaster 37 which runs on various hardware architectures including GPUs and high memory multi core CPU architectures. We make use of the large CPU memory to benchmark the convergence of the hierarchy depth and use the high compute throughput of the GPUs for production runs.
In Section 2 we outline the structure of the Frenkel exction model for energy transfer in CSM. The technical aspects of the HEOM approach are stated in Section 3. After that, we continue with the discussion of time-scales of inter-protein transfer in the PS-II supercomplex (see Section 4). Finally, we investigate the impact of structural modifications on the transfer pathways and the transfer efficiency.
2 Exciton Model
The orientation of the individual proteins of the CSM supercomplex is determined by a projection map at 12 Å resolution 5. The CSM supercomplex, which structure is depicted in Fig. 1(a), comprises four LHCII trimers, six minor light-harvesting complexes and a dimeric PS-II core complex. Absorbed light in the outer LHCII antenna complexes is transfered via the minor complexes CP24, CP26, and CP29 to CP47 and CP43 of PS-II. The transfer process is completed by irreversible charge separation triggered in the RC core. Electron transfer in the RC core is described phenomenologically by radical pair states RP1 , RP2 and RP3 39, 12. We assume that the primary step of charge separation is initiated through the electronically excited core pigment Chl (the location of Chl in PS-II is illustrated in Fig. 1(b)) and described by the rate equation
We neglect backward rates since fluorescence decay lines suggest that charge recombination occurs on a much slower time scale than primary electron transfer 12. Within this limit we model primary charge separation as irreversible exciton trapping. In literature also more sophisticated models are discussed which include multiple pathways of charge separation 40, 41.
We describe energy transfer in the CSM supercomplex within a Frenkel exciton Hamiltonian for which we assume that only one of the pigments is excited at once. The Hamiltonian of the single exciton manifold reads
Here denotes the state in which pigment is excited while the other pigments remain in the electronic ground state. For the inter-site couplings we distinguish between intra-complex and inter-complex coupling terms depending whether or not pigments and are located within the same protein. We use the same parameter for the exciton Hamiltonian which is constructed by Bennett in Ref. 22. Recently the Hamiltonian for CP29 has been resolved 15 which is in Ref. 22 replaced by a LCHII monomer (without pigment Chl 605). In order to isolate of how much the new CP29 Hamiltonian influences transfer and to compare the HEOM results with previous approximate modified Redfield/generalized Förster simulations 22, we carry out calculations for both models: (i) with the CP29 Hamiltonian and (ii) with the LHCII monomer substitution.
The pigments are coupled to the protein environment modeled by a set of independent harmonic oscillators
and we assume a linear coupling of the exciton system to the vibrations
The reorganization energy , with is added to the exciton Hamiltonian eq. (2). We define the site energies as . The phonon mode dependent coupling strength is captured by the spectral density
Frequently, the reorganization energy and the spectral density are assumed to be site independent. However for the CSM supercomplex each of individual protein has its own reorganization energies and own form for the spectral density 22. The spectral density for LHCII is extracted from fluorescence line narrowing spectra. Since the experimental spectra cannot distinguish between the Chl and Chl pigments we assume for both the same spectral density composed of 48 vibrational peaks 42, 43. Transfer times are not much affected by the structures in the spectral density and a coarse grained Drude-Lorentz spectral density is appropriate to describe energy transfer in LHCII 37. Microscopic details for the spectral densities of the minor complexes and PS-II are not know. The structure of CP29 is similar to the one of a LHCII monomer. Thus we assume that the spectral density of CP29 can be substituted with the LHCII spectral density22, 15. For CP47 and the RC core pigments cm and cm, respectively are suggested as reasonable values for the reorganization energy 12. For the RC core pigments also a higher reorganization energy is discussed 40, 44. The explicit form and parameter for the spectral densities used in this manuscript are listed in Appendix A.
We include the primary step of charge separation phenomenologically as irreversible population trapping, which we incorporate by anti-Hermitian parts in the Hamiltonian
where defines the rate of the primary charge separation. In a similar way we incorporate exciton losses
where we assume exciton lifetimes of ns. We characterize transfer properties by the transfer efficiency
and average transfer time
For numerical evaluations we replace the upper integration limit by which is chosen such that the total population within the pigments of the CSM supercomplex has dropped below .
We evaluate the exciton dynamics within the hierarchically coupled equation of motion (HEOM) method 30, 33, 34. HEOM is an open quantum system approach which treats the coupling to the vibrational modes as a bath. The time evolution of the total density operator , which characterizes the degrees of freedom of the exciton system as well as the ones of the phonon bath, is governed by the Liouville equation
We assume that the total density operator factorizes at initial time in system and vibrational degrees of freedom . To get the time evolution of the reduced density matrix , we trace out the bath degrees of freedom
By employing second order cumulant expansion, using a Drude-Lorentz spectral density in combination with a high temperature approximation , we cast the time non-local eq. (11) into a hierarchy of coupled time local equations of motion
where we define , , , and . The hierarchy can be truncated for a sufficiently large hierarchy level . Convergence of the hierarchy can be tested by comparing deviations in the dynamics with increasing truncation level. To increase the accuracy of the high temperature approximation of HEOM 45 we include additional correction terms 46 for which we replace
For structured spectral densities a similar hierarchical expansion has been derived which relies on a decomposition of the spectral density in terms of shifted Drude-Lorentz peaks 47, 48 or underdamped Brownian oscillators 34.
Together with LHCII complexes, the CSM supercomplex aggregates as a large photosynthetic network in the grana membrane. For each CSM supercomplex there are about six additionally loosely bound LHCII trimers 4, which form a large antenna system with densely packed chlorophylls. Energy is either absorbed in the pool of loosely bound LHCII trimers and then transfered to one of the peripherical LHCIIs of the CSM supercomplex or absorbed directly in the LHCII trimers of CSM. Further, to some extend energy is absorbed in the minor complexes and PS-II. We expect that the contribution of light absorption in the minor complexes and PS-II to the photosynthetic yield is of less importance, since most of the photoactive area in the grana membrane is covered by the LHCIIs. Thus, to reach a high photosynthetic yield fast and efficient transfer from the LHCIIs toward the RC core pigments of PS-II becomes indispensable.
In the following we investigate average transfer times and the efficiency of energy transfer from the peripherical LHCII-m monomeric unit labeled as unit-1 in Fig. 1 to the reaction center in which the primary step of charge separation takes place. We employ the presence of a certain amount of symmetry along x-axis and y-axis and reduce the system to a multi-protein network composed of LHCII-m, CP24, CP29, CP47 and the RC-core, comprising 93 pigments in total.
4.1 Energy gradient drives directionality
First, we keep track of the population dynamics in absence of trapping and energy losses. We highlight of how energy spreads among the different protein complexes which, as we will analyze in detail, is driven by energy gradients in the pigments of the CSM supercomplex. Further, we explore the influence of the reorganization process on the exciton dynamics.
Following, we investigate the deficiency of the approximate combined modified Redfield/generalized Förster method by comparing the population dynamics obtained from the combined method with the HEOM results. The combined modified Redfield/generalized Förster approach divides the exciton Hamiltonian eq. (2) into a strongly coupled part (associated with strongly coupled domains) and a weakly coupled part. Hereby is defined as the exciton Hamiltonian in which the inter-site couplings are set to zero if the coupling strength drops below a certain threshold value. We follow Ref. 22 and use a threshold of 15 cm. The intra domain dynamics is then modeled by modified Redfield, while the inter-domain transfer is described by generalized Förster theory. Since the choice of initial conditions of the combined method is restricted to eigenstates of certain domains in we set the highest energy state of the domain which predominantly populates pigment Chl 606 of the LHCII-m unit-1 as initial condition. To allow for comparison, we use the same initial condition for the HEOM calculations.
Figure 2(a) depicts the aggregated population at the individual protein complexes obtained within HEOM. Convergence of the hierarchy depth is verified by comparing with a higher truncation level (see Appendix B). Overall energy transfer and directionality is driven by energy relaxation along energy gradients within the CSM supercomplex. LHCII and the minor complexes (modeled by LHCII monomers without Chl 605) exhibit the highest energies while CP47 and the RC core pigments are lower in energy. The exciton, initially located at the unit-1 LHCII-m monomer, spreads over the complete LHCII-m trimer and populates the minor complexes. The fast initial spread, shows as maxima in the aggregated populations at LHCII-m units-2 and 3. The highest population at the unit-3 is obtained after about 18 ps while the maximum population at unit-2 is reached a bit later at about 43 ps. The high population of the LHCII-m unit-2 indicates that energy transfer does not exclusively proceed along pathways corresponding to the shortest distance to PS-II and the RC. The minor complexes are populated on a similar time-scale as the monomeric LHCII-m units. On a slower timescale CP47 and the RC core of PS-II get populated, and finally after about 250 ps - 350 ps the system reaches steady state in which energy relaxation drives the population to the energetically low exciton states at CP47. A schematic sketch summarizing the rough estimates for the energy transfer time-scales is given in Fig. 2(d).
The dynamics of the combined modified Redfield/generalized Förster approach (Fig. 2(b)) diverges from the HEOM results in several aspects. Overall relaxation is overestimated by the combined modified Redfield/generalized Förster approach. Especially transfer to LHCII-m unit-2 is about seven times faster and already at 6 ps there is 0.29 population at the unit-2. Further, unit-2 gets populated ahead of unit-3. Therefore the pathway of how energy spreads over the monomeric units of the LHCII-m trimer is reversed when compared to the HEOM calculation and thus the combined method does not predict reliable pathways of energy flow during the first picoseconds. However, the main difference is in the stationary population which is not only approached faster (at about 150 ps - 250 ps) but predicts a much higher aggregated population at CP47 and the RC. The combined modified Redfield/generalized Förster approach overestimates the efficiency of energy funneling towards the PS-II.
To understand the discrepancy in the stationary population we need to investigate the energetic layout of the CSM supercomplex. The stationary state follows typically a thermal Boltzmann distribution. However, the situation becomes more complicated in presence of the reorganization process in which the reorganization energy dissipates during the dynamics which modifies the energetic layout and affects the thermal population. The boxes in Fig. 2(c) indicate the extension of the exciton bands of the isolated proteins. The dashed lines correspond to the situation where the site energies comprise of the bare excitation energy plus the reorganization energy. Due to the reorganization process the energetic structure changes during the dynamics and the energy of the proteins is lowered by the reorganization energy. Especially the band of the monomeric LHCII-m units and the band of the minor complexes shifts to lower energies while the small reorganization energies at CP47 and RC induce only minor modifications. In total the already flat energy gradient gets even more flattened. This has a significant impact on the thermal population. Without the reorganization process (dashed lines) we expect a thermal population of about 0.75 at the pigments of PS-II. Taking into account the reorganization process (solid line) reduces the efficiency of energy funneling and only a population of 0.56 accumulates at PS-II. Our analysis is in consistency with the findings for the population dynamics and explains the strong deviations in the stationary state between HEOM and the combined modified Redfield/generalized Förster method. We note that for the combined modified Redfield/generalized Förster method the effect of reorganization energy on the thermal population can be corrected by subtracting the reorganization energy from the exciton Hamiltonian prior to the dynamics. This is based on the assumption that the reorganization energy dissipates on an infinitely fast time-scale.
4.2 Structural variations modify the transfer efficiency
In the following we investigate how minor structural modifications in the CSM supercomplex influence transfer properties such as transfer efficiency and average transfer time. As we have discussed in detail in the previous section, one mechanism that induces structural changes is the reorganization process. Here, we continue the discussion and examine how much the reorganization process affects transfer efficiency. Another aspect is the influence of the replacement of the minor complexes with LHCII monomeric units on the transfer properties. For instance the recently derived exciton Hamiltonian of CP29 shows various differences from the exciton system of a LHCII monomer 15.
We incorporate the primary step of charge separation by irreversible energy trapping as is described in section 2. Different values for the rate constant of primary charge separation have been predicted from fits to fluorescence decay lines, ranging from =0.1 ps 12 to =0.64 ps 22. Pump-probe spectra predict even larger time constants for the Pheo reduction of about 3 ps 39. We do not explicitly take into account mechanisms of photoprotection and quenching and phenomenologically describe exciton losses by assuming an exciton lifetime of ns.
In the following we carry out HEOM simulations in which we include trapping and energy losses. To investigate the effects of the reorganization process on the energy transfer times, we slightly modify the Hamiltonian of the CSM supercomplex in a benchmark calculation for which we artificially restore the original energy gradient across the pigment proteins by adding the reorganization energy of 220 cm to the site energies of LHCII and the minor complexes. We neglect the minor energetic changes induced by the reorganization process at the pigments of CP47 and the RC and denote the modified Hamiltonian as . Relaxation time scales in the population dynamics are hardly affected by the shifts in the site-energies, but the thermal state adjusts now according to the modified energy gradient. For we obtain a similar thermal state in the population dynamics with high population at the PS-II pigments (0.81) as is predicted by the calculations with the combined modified Redfield/generalized Förster method. The small deviations largely result from the fact that we did not add additional reorganization energies to the site energies of CP47 and RC.
The transfer time as function of trapping rate follows a linear trend for the considered parameter regime as is illustrated in Fig. 3. We assume that initially the exciton is located at the LHCII-m unit-1, and we populate the initial density matrix according to eigenstates of the isolated LHCII monomeric unit. The shown results correspond to transfer times averaged over all 14 exciton states used as initial condition. Transfer times (efficiency) for the CSM supercomplex (marked by the red circles) are in the range between 242 ps (88.0%) and 302 ps (85.2%), depending on the trapping rate . exhibits a more efficient energy energy funneling towards the pigments at PS-II and therefore transfer is faster by about 36 ps to 53 ps. Previous calculations based on the combined modified Redfield/generalized Förster method predict transfer times of about 200 ps for transfer from peripherical domains to the Chl D1 in the RC 22. This is in good agreement with our results for which yields a transfer time of 211 ps for trapping rate of ps which is similar to the one used in Ref. 22.
4.3 Minor complex CP29 guides transfer
Since already small structural modifications such as the reorganization process alters the energy transfer efficiency, we expect that the substitution of the minor complexes by LHCII monomers may also significantly affect the energy transfer properties. In this section we use the Hamiltonian of CP29 derived by Renger Ref. 15 instead of the LHCII monomer replacement. We denote the new Hamiltonian as , while the previous situation with the LHCII monomer substitution is referred to as .
For the pigments of CP29 form the lowest energy band in the energetic layout of the CSM supercomplex. This has a two-fold implication on the transfer process. Firstly, the energy gradient between the outer LHCII antenna and the minor complex CP29 gives rise to an additional grade of directionality and supports fast transfer from the peripherical LHCII-m trimer to CP29. The minor complex CP29 presumably acts as exit marker which guides energy from the outer antenna towards pigments closer to the reaction center. Secondly, the pigments of CP29 and CP47 form a spatially extended region of low-energy states and hence energy accumulates at pigments in proximity to the RC, while the final transfer step to the RC core pigments is energetically uphill and therefore slow. Overall the two effects result in a slightly slower energy transfer within the CSM supercomplex while including the CP29 Hamiltonian, see Fig. 3. For large trapping rates ps the slow down of the energy transfer gets more pronounced.
Figure 4 charts snapshots of the exciton dynamics. The upper (lower) panels correspond to (). The radius of the colored spheres represents the population at each pigment. For better visualization we use an arctan scale. The spheres are uncolored if the population remains below 0.0079. Initially the highest eigenstate of the LHCII-m unit-1 is excited. Both Hamiltonians show a fast spread of the energy and at 12 ps the energy distributes across the whole LHCII-m timer. While distributes population equally among the minor complexes yields a more directed energy transfer towards the CP29 and CP47. For longer times of 150 ps energy accumulates at the low energy states at CP29 and CP47 for and thus forms a bottleneck for transfer to the RC. The bottleneck is less pronounced for . The RC pigments do not show significant population at any time since as soon as energy enters the RC there is fast transfer to Chl D1 and the fast time-scale of primary charge separation leads to the trapping of the population.
The rate limiting step in the transfer chain is the energetically up-hill transfer to the RC core. This is illustrated best in the aggregated population dynamics in presence of trapping in the reaction center, Fig. 5. We obtain a fast decay of population in the LHCII-m and after 100 ps more than 0.75 of the population has left the LHCII-m trimer. At the same time about 0.44 of population accumulates in CP29 and CP47. After 300 ps still 0.2 of the population remains at CP29 and CP47.
With QMaster, a high-performance implementation of the HEOM method, accurate calculations of excitonic energy transfer in multi-protein photosynthetic functional units such as the CSM supercomplex become feasible. We investigate transfer times and transfer efficiency of energy conversion within the primary step of charge separation.
The general concept behind energy transfer in CSM is given by energy relaxation. Due to the flat energy gradient across the proteins, small structural changes such as the reorganization of the molecular coordinates within the excited potential energy surface, affect the energy transfer process. The impact of the reorganization process is rather significant and energy relaxation drives much less population to CP47 and the RC than expected from the site energies of the Hamiltonian. The reorganization process induces a noticeable drop in the transfer efficiency of about 1.8% to 2.6% in absolute numbers for a 2 ps exciton lifetime, and thus cannot be neglected in simulations of energy transfer in large multi-protein complexes.
Our simulations suggest that the minor complex CP29 acts as exit marker and adds directionality to the energy transfer from the peripherical LHCII to the proteins in the proximity to the RC core. The CSM supercomplex is not optimized for efficient transfer. Energy accumulates in low energy states at CP29 and CP47, while the final transfer step needs to overcome an energy barrier and therefore is slow. Thus the energy transfer exhibits the character of a transfer-to trap limited model. In conclusion, within our model, we show that the structural layout of CSM is not optimized for efficient transfer and suggests that photoprotection considerations are very relevant. The extension of accurate HEOM models to this case is possible and a promising direction for future research.
The authors would like to thank Dr. Kapil Amernath and Dr. Doran Bennett for helpful discussion. We thank Dr. Doran Bennett for providing the atomistic structure and the Hamiltonian of CSM. This work was supported as part of the Center for Excitonics, an Energy Frontier Research Center funded by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001088. We thank Nvidia for support via the Harvard CUDA Center of Excellence. The computations in this paper were run on the Odyssey cluster supported by the FAS Division of Science, Research Computing Group at Harvard University.
Appendix A Parameter of the spectral densities
The coupling of each pigments to the vibrational environment is described by a Drude-Lorentz spectral density
The paramter and for the individual pigments are listed in Table 1.
|LHCII, CP29 and CP24||220 cm||15 fs|
|CP47||38.64 cm||50 fs|
|RC core||50.23 cm||50 fs|
Appendix B Convergence of HEOM
In order to test convergence of HEOM the dynamics is compared for different truncation levels as is charted in Fig. 6. Initially the highest exciton state of LHCII-m unit 1 monomer is excited. A truncation level of yields sufficient accuracy for the relaxation time-scales.